Phase reconstruction from oscillatory data with iterated Hilbert transform embeddings -- benefits and limitations

TL;DR

This paper evaluates the iterated Hilbert transform embeddings method for phase reconstruction in oscillatory data, highlighting its benefits under certain conditions and its limitations with amplitude modulation.

Contribution

It introduces a practical procedure for phase reconstruction using iterated Hilbert transform embeddings and assesses its effectiveness on a forced Stuart-Landau oscillator.

Findings

Improves phase reconstruction with strong oscillation stability and high forcing frequency.

Fails to improve phase reconstruction with significant amplitude modulation.

Highlights limitations of the method in practical scenarios.

Abstract

In the data analysis of oscillatory systems, methods based on phase reconstruction are widely used to characterize phase-locking properties and inferring the phase dynamics. The main component in these studies is an extraction of the phase from a time series of an oscillating scalar observable. We discuss a practical procedure of phase reconstruction by virtue of a recently proposed method termed \textit{iterated Hilbert transform embeddings}. We exemplify the potential benefits and limitations of the approach by applying it to a generic observable of a forced Stuart-Landau oscillator. Although in many cases, unavoidable amplitude modulation of the observed signal does not allow for perfect phase reconstruction, in cases of strong stability of oscillations and a high frequency of the forcing, iterated Hilbert transform embeddings significantly improve the quality of the reconstructed phase. We also demonstrate that for significant amplitude modulation, iterated embeddings do not provide any improvement.